Which of the following interchanges of signs and numbers will make the equation 8 × 9 + 2 = 74 correct, when standard arithmetic rules are applied?

Introduction / Context:
This question tests your ability to manipulate an equation by interchanging both operation signs and numbers so that the resulting statement becomes arithmetically correct. You are given several possible pairs of items to swap, and you must check which particular interchange yields a true equation under normal arithmetic rules. It combines algebraic thinking with logical trial and error.

Given Data / Assumptions:

- The original equation is: 8 × 9 + 2 = 74.
- Each option suggests swapping two signs and also swapping two numbers.
- Only one option should make the resulting equation numerically correct.
- After the swap, we use the usual operations and standard precedence (multiplication before addition).

Concept / Approach:
For each option, we conceptually perform two swaps:

- First, swap the specified numbers everywhere they appear.
- Second, swap the specified signs everywhere they appear.

After performing both swaps, we evaluate the resulting equation. If the left-hand side equals the right-hand side under standard arithmetic, that option is valid; otherwise, it is rejected. We repeat this until we find the single correct option.

Step-by-Step Solution:
Original equation: 8 × 9 + 2 = 74. Check option B: swap "× and +" and swap "8 and 2". Step 1: Swap the numbers 8 and 2. The equation becomes: 2 × 9 + 8 = 74. Step 2: Swap the signs × and +. Every "×" becomes "+", and every "+" becomes "×". The equation becomes: 2 + 9 × 8 = 74. Step 3: Evaluate using precedence. First compute 9 × 8 = 72. Then compute 2 + 72 = 74. The equation 2 + 9 × 8 = 74 is true, so option B works. For completeness, other options either violate the equation or produce noninteger equality. For example, option A leads to 8 = 2 + 9 × 74, which is clearly false.

Verification / Alternative check:
We can quickly cross check option B by redoing the swaps more compactly: writing the final equation directly as 2 + 9 × 8 = 74 and verifying that the arithmetic is correct. Since 9 × 8 is 72 and adding 2 gives exactly 74, this confirms the correctness without ambiguity.

Why Other Options Are Wrong:
Option A results in an equation of the form 8 = 2 + 9 × 74, where the right-hand side is far larger than 8. Option C gives 9 + 8 × 2 = 74, which evaluates to 9 + 16 = 25, not 74. Option D reorganizes the right-hand side but does not satisfy equality either. Hence none of these alternatives balance the equation, leaving option B as the only correct choice.

Common Pitfalls:
A common mistake is to perform only one of the swaps, or to swap numbers correctly but forget to change all instances of the signs. Another frequent slip is to ignore operator precedence after the swap and evaluate from left to right, leading to an incorrect check of the resulting equation. Careful, stepwise transformation followed by correct evaluation avoids these errors.

Final Answer:
The equation becomes correct if we interchange "× and +" and swap the numbers 8 and 2. Therefore, the correct choice is × and +, 8 and 2.